Concise Physical Chemistry

c16 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

BREAKING DEGENERACY 259

Upon writing this equation out in full, it becomes rather intimidating, as the

reader will see by scanning several textbooks. Fortunately, expressed inr,θ,andφ,

it breaks up into three separate equations just as the particle in a cubic box did. The

equations are

∂

∂r

r^2

∂R(r)

∂r

+

2 mer^2

̄h^2

(

e^2

r

+E

)

R(r)=R(r)β

1

sinθ

∂

∂θ

sinθ

∂   (θ)

∂θ

−

m^2 e

sin^2 θ

=−β (θ)

and

(φ)=−

1

√

2 π

eimeφ

Each of these equations involves only one variableR(r), (θ),and (φ). The first

R(r) is called theradial equationand the second two are often lumped together

as thespherical harmonics Y(θ,φ)= (θ) (φ). The name is apt; they describe

oscillations taking place on the surface of a sphere with the boundary constraints that

no wave can have a discontinuity or seam anywhere on the sphere.

The wave function or orbital for the electron in the ground state of the hydrogen

atom is(r)=e−αr, whereris the radial distance between the proton and the

electron, andαcontains several constants. Solution forαby any one of several

methods gives

E=−

1

2

(

me^4

( 4 πε 0 )^2 ̄h^2

)

=−

me^4

32 π^2 ε^20 h ̄^2

This purely quantum mechanical result is in precise quantitative agreement with

Bohr’s semiclassical result for the ground state of hydrogen. When higher energy

states are considered, quantum numbers appear just as they do in Bohr’s result.

In addition to the principal quantum numbern, a more general solution for the

spherical harmonicsY(θ,φ)= (θ) (φ) leads to a quantum numberlfor the (θ)

equation andmfor the (φ) equation. There are also restrictions on the quantum

numbers in the complete solution. For example,n≤l+1, so thatlcan take on a

zero value even thoughncannot.

16.8 BREAKING DEGENERACY

For the simple one-electron system of hydrogen, the 2sand 2porbitals are degenerate,

as are the more complicated 3s,3p,and 3dorbitals. When we start putting electrons

into these orbitals tobuild uptheatomictablebytheaufbauprinciple, however,

some of these degeneracies are lost. For example, the probability density function

of the 2porbital is small near the nucleus, where, in contrast, thesorbital has a